Probing the emission mechanism and nature of the pulsating compact object in the X-ray binary SAX J1324.4-6200

Recently, there has been renewed interest in the Be X-ray binary (Be/XRB) SAX J1324.4-6200 because of its spatial coincidence with a gamma-ray source detected by Fermi/LAT. To explore more thoroughly its properties, new observations were carried out in 2023 by NuSTAR, XMM-Newton, and Swift, jointly covering the energy range 0.2-79 keV. The X-ray spectrum of SAX J1324.4-6200 fits well with an absorbed power law with a high energy cut-off. We measured a NuSTAR spin period of 175.8127 +/- 0.0036 s and an XMM-Newton spin period of 175.862 +/- 0.025 s. All the available spin period measurements of SAX J1324.4-6200, spanning 29 years, are correlated with time, resulting in a remarkably stable spin-down of dP/dt=(6.09 +/- 0.06)*1E-9 s/s. If SAX J1324.4-6200 hosts an accretion powered pulsar, accretion torque models indicate a surface magnetic field of  1E12-1E13 G. The X-ray properties emerging from our analysis strenghten the hypothesis that SAX J1324.4-6200 belongs to the small group of persistent Be/XRBs. We also performed radio observations with the Parkes Murriyang telescope, to search for radio pulsations. However, no radio pulsations compatible with the rotational ephemeris of SAX J1324.4-6200 were detected. We rule out the hypothesis that SAX J1324.4-6200 is a gamma-ray binary where the emission is produced by interactions between the pulsar and the companion winds. Other models commonly used to account for the production of gamma-rays in accreting pulsars cannot reproduce the bright emission from SAX J1324.4-6200. We examined other mechanisms for the gamma-ray emission and noted that there is a  0.5 probability that an unknown extragalactic AGN observed through the Galactic plane may coincidentally fall within the Fermi/LAT error circle of the source and be the responsible of the gamma-ray emission. [Abridged]